Idea

Ben Webster: Perhaps something could be said about an extended TQFT FF? My understanding was that the decategorification of F(X)F(X) was given by F(X×S1)F(X\times S^1); is this right?

Urs Schreiber: that process certainly makes an nn-dimensional QFT becomes an (n−1)(n-1)-dimensional one. It is pretty much exactly the mechanism of fiber integration.

So this certainly does have a flavor of decategorification. But the latter also has a precise sense in terms of taking equivalence classes in a category. So at face value fiber integration is something different. But perhaps there is some change of perspective that allows to regard it as decategorification in the systematic sense.

Extra structure

If the category in question has extra structure, then this is usually inherited in some decategorified form by its decategorification. For instance if CC is a monoidal category then K(C)K(C) is a monoid.